To the best of the author's knowledge is a more formal way to say as far as he or she (the author) knows. Say that one has written down something in a scientific publication. He or she has thoroughly searched the relevant literature but he or she could not find a single reference of the result in question.

The author has searched carefully but he or she cannot searched the WHOLE literature of an enormous field like, e.g. continuum mechanics. So one uses this expression. I am not a French native speaker but I thought it should not sound so awkward in French.

As a real example (taken from Freed's book: Soft Solids):

"The Eulerian velocity gradient l is typically denoted as L in the literature. That notation is reserved here for the Lagrangian velocity
gradient defined in Eq. (2.24), which is a field not found in the
literature to the best of the author’s knowledge."

@SteffX Thus, if I understood correctly, there is not a direct French interpretation of "to the best of the author's knowledge" ? One should use the personal structure "à ma connaissance" ?
– DimitrisDec 22 '18 at 17:04

— That's right! The "best of ... knowledge" had an equivalent in French (which I can't remember now) but is no longer used. Maybe we considered that "to the best of author's knowledge" is a bit weird. What was the author doing before? Not using his "best knowledge"?
– SteffXDec 22 '18 at 17:41

@SteffX To the best of the author's knowledge is a more formal way to say as far as he (the author) knows. Say that you write down something in a publication. You have searched in the literature but you could no find a single reference of your result. You have searched carefully but you cannot searched the WHOLE literature of an enormous field like, e.g. continuum mechanics. So one uses this expression. I am not a French native speaker but I think it should not sound weird.
– DimitrisDec 22 '18 at 17:48

@SteffX See a real example: (taken from Freed: Soft Solids) "The Eulerian velocity gradient l is typically denoted as L in the literature. That notation is reserved here for the Lagrangian velocity gradient defined in Eq. (2.24), which is a field not found in the literature to the best of the author’s knowledge.
– DimitrisDec 22 '18 at 17:51